Lemma: Root composition

The composition (eq:B-kernel) exists, has compact support, and satisfies

Its scalar projection is contained in $O_{f,g}$.

Proof. The $z$ support of $A$ is compact. A forbidden product cancellation would have zero covectors in the external variables $y,x$. A Green wavefront covector has the form $(p,-p)$, so the zero $y$ covector forces that Green covector to vanish. Then (eq:A-WF) and the zero $x$ covector force the root covector to vanish as well. The product is therefore defined.

The $z$ projection is proper on the relevant support. For a pushforward wavefront point, cancellation of the internal $z$ covectors leaves output covectors $(p,-p)$. This proves the bound. The time collar makes the support compact; the scalar projection follows directly from (eq:A-kernels).